The complexity of topological group isomorphism

THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM

arXiv:1705.08081v1 [math.LO] 23 May 2017

´ NIES AND KATRIN TENT ALEXANDER S. KECHRIS, ANDRE

Abstract. We study the complexity of the isomorphism relation for various classes of closed subgroups of S∞ . We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

1. Introduction Let S∞ denote the Polish group of permutations of ω. It is well-known that the closed subgroups of S∞ (or equivalently, the non-Archimedean Polish groups) are, up to topological group isomorphism, the automorphism groups of countable structures. Algebra or model theory can sometimes be used to understand natural classes of closed subgroups of S∞ . Firstly, the separable profinite groups are precisely the Galois groups of Galois extensions of countable fields. For a second example, consider the oligomorphic groups, namely the closed subgroups of S∞ such that for each n there are only finitely many n-orbits. They are precisely the automorphism groups of ω-categorical structures. Under this correspondence, topological isomorphism turns into bi–interpretability of the structures by a result of Ahlbrandt and Ziegler [1] sometimes also attributed to Coquand. The closed subgroups of S∞ form the points of a standard Polish space. Our main goal is to determine the complexity of the topological isomorphism relation for various classes of closed subgroups of S∞ within the setting of Borel reducibility between equivalence relations. See [4] for background on this setting. An important question about an equivalence relation E on a standard Borel space X is whether E is classifiable by countable structures. This means that one can in a Borel way assign to x ∈ X a countable structure Mx in a fixed countable language so that xEy ⇔ Mx ∼ = My . We consider the Borel classes of compact (i.e., profinite) groups, locally compact groups, and oligomorphic groups. We also include the class of Roelcke precompact groups, which generalise both the compact and the oligomorphic groups. We introduce a general criterion on a class of closed subgroups of S∞ to show that each of the classes above has an isomorphism relation that is classifiable by countable structures. Our proof that the criterion works has two different versions. The first is on the descriptive set theory side: we Borel reduce the isomorphism relation to conjugacy of closed subgroups of S∞ , which implies classifiability by countable structures using a result The first author was partially supported by NSF grant DMS 1464475. The second author was partially supported by the Marsden fund of New Zealand. The third author was supported by Sonderforschungsbereich 878 at Universit¨ at M¨ unster. 1

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due to Becker and Kechris [2, Thm. 2.7.3]. The second is on the model theoretic side: we directly construct a countable structure MG in a fixed finite language from a group G in the class so that topological isomorphism of groups is equivalent to isomorphism of the associated structures. Independently from us, Rosendal and Zielinski [10, Prop. 10 and 11] established classifiability by countable structures for the isomorphism relation in the four classes above, and published their result on arXiv in Oct. 2016. Their methods are different from ours: they obtain the results as corollaries to their theory of classification by compact metric structures. We conversely provide a Borel reduction of graph isomorphism to isomorphism of profinite groups, using an extension of an argument by Mekler [7] in the framework of topological groups. In fact, for p an odd prime, the class of exponent p, nilpotent of class 2, profinite groups suffices. For isomorphism of oligomorphic groups, it is clear that the identity on R is a lower bound (e.g. using Henson digraphs); we leave open the question whether this lower bound can be improved. We note that for ω-categorical structures in a finite language, the bi-interpretability relation is Borel and has countable equivalence classes. Since graph isomorphism is not Borel, this upper bound for the isomorphism relation of the corresponding automorphism groups is not sharp. Using Lemma 2.1 below, it is not hard to verify that the isomorphism relation for general closed subgroups of S∞ is analytical. It is unknown what the exact complexity of this relation is in terms of Borel reducibility. By the above, graph isomorphism is a lower bound. 2. Preliminaries Effros structure of a Polish space. Given a Polish space X, let F(X) denote the set of closed subsets of X. The Effros structure on X is the Borel space consisting of F(X) together with the σ-algebra generated by the sets CU = {D ∈ F(X) : D ∩ U 6= ∅}, for open U ⊆ X. Clearly it suffices to take all the sets U in a countable basis hUi ii∈ω of X. The inclusion relation on F(X) is Borel because for C, D ∈ F(X) we have C ⊆ D ↔ ∀i ∈ N [C ∩ Ui 6= ∅ → D ∩ Ui 6= ∅]. The following fact will be used frequently. Lemma 2.1 (see [6], Thm. 12.13). Given a Polish space X, there is a Borel map f : F(X) −→ X ω such that for a non-empty set G ∈ F(X), the image ω f (G) is a sequence (pG i )i∈ω in X that is dense in G. The Effros structure of S∞ . For a Polish group G, we have a Borel action G y F(G) given by left translation. In this paper we will only consider the case that G = S∞ . In the following σ, τ, ρ denote injective maps on initial segments of the integers, that is, tuples of integers without repetitions. Let [σ] denote the set of permutations extending σ: [σ] = {f ∈ S∞ : σ ≺ f } (this is often denoted Nσ in the literature). The sets [σ] form a base of S∞ . For f ∈ S∞ let f ↾n be the initial segment of f of length n. Note that the [f ↾n ] form a basis of neighbourhoods of f . Given σ, σ ′ let σ ′ ◦ σ be the composition as far as it is defined; for instance, (7, 4, 3, 1, 0)◦(3, 4, 6) = (1, 0i.


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Definition 2.2. For n ≥ 0, let τn denote the function τ defined on {0, . . . , n} such that τ (i) = i for each i ≤ n. Definition 2.3. For P ∈ F(S∞ ), by TP we denote the tree describing P as a closed set in the sense that [TP ] ∩ S∞ = P . Note that TP = {σ : P ∈ C[σ] }. Lemma 2.4. The relation {(A, B, C) : AB ⊆ C} on F(S∞ ) is Borel. Proof. AB ⊆ C is equivalent to the Borel condition ∀β ∈ TB ∀α ∈ TA [|α| > max β → α ◦ β ∈ TC ]. For the nontrivial implication, suppose the condition holds. Given f ∈ A, g ∈ B and n ∈ N, let β = g ↾n , and α = f ↾1+max β . Since α ◦ β ∈ TC , the neighbourhood [f ◦ g ↾n ] intersects C. As C is closed and n was arbitrary, we conclude that f ◦ g ∈ C. The Borel space of non-Archimedean groups. Lemma 2.5. The closed subgroups of S∞ form a Borel set U (S∞ ) in F(S∞ ). Proof. D ∈ F(S∞ ) is a subgroup iff the following three conditions hold: • D ∈ C[(0,1,...,n−1i] for each n • D ∈ C[σ] → D ∈ C[σ−1 ] for all σ • D ∈ C[σ] ∩ C[τ ] → D ∈ C[τ ◦σ] for all σ, τ . It now suffices to observe that all three conditions are Borel. Note that U (S∞ ) is a standard Borel space. The statement of the lemma actually holds for each Polish group in place of S∞ . Locally compact closed subgroups of S∞ . These groups are exactly the (separable) totally disconnected locally compact groups. The class of such groups has been widely studied. A set D ∈ F(S∞ ) is compact iff the tree TD = {σ : D ∈ C[σ] } is finite at each level. A closed subgroup G of S∞ is locally compact iff some point in G has a compact neighbourhood. Equivalently, there is τ such that G ∈ C[τ ] and the tree {σ τ : G ∈ C[σ] } is finite at each level. Thus, compactness and local compactness of subgroups are Borel conditions in F(S∞ ). The canonical structure for a closed subgroup of S∞ . Given G ∈ U (S∞ ) we can in a Borel way obtain a countable structure MG in a countable signature such that G ∼ = Aut(MG ). For each n, order the n-tuples lexicographically. Let hai ii 0, let Un ∈ NG be such that Un ≤ [τn ] ∩ Un−1 , where as above τn is the identity tuple of length n + 1. Define g(n) = rn (n) where rn ∈ G and rn Un is the left coset of Un in L. Also define g ∗ (n) = s−1 n (n) where Un sn is the right coset of Un in R.

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Clearly the rn ↾n+1 (n ∈ ω) are compatible tuples with union g. Similarly ∗ the s−1 n ↾n+1 are compatible tuples with union g . However, so far we don’t know that the injective function g is a permutation. Claim 3.6. g∗ is the inverse of g. In particular, g ∈ S∞ . Proof. Note that rn determines the first n + 1 values of any permutation f in rn Un , namely f (i) = rn (i) for each i ≤ n. Likewise, h ∈ Un sn implies −1 −1 that h−1 ∈ s−1 n Un and hence h (i) = sn (i) for each i ≤ n. Assume for a contradiction that, say g(x) = y and g∗ (y) 6= x. Let n = max(x, y). Then rn Un ∩ Un sn = ∅, contrary to (2) above: if f ∈ rn Un ∩ Un sn then by the compatibility and the observation above, f (x) = rn (x) = y, while f −1 (y) = s−1 n (y) 6= x. Claim 3.7. (i) g ∈ G. (ii) Lg = L and Rg = R. Proof. (i) follows because g ↾n+1 = rn ↾n+1 and G is closed. (ii) is clear from the definition of g and g ∗ = g−1 . Claim 3.8. For a, b, c ∈ G, a ◦ b = c ↔ ∀C ∈ Lc ∃A ∈ La ∃B ∈ Lb [AB ⊆ C]. Proof. The implication from left to right holds by continuity of composition in G. For the converse implication, suppose that (a ◦ b)(n) 6= c(n). Let C ∈ Lc be the left coset of Un . Then d(n) = c(n) for any d ∈ C. If A ∈ La and B ∈ Lb then a ◦ b ∈ AB so that AB 6⊆ C. Now suppose that MG ∼ = MH via ρ. Given g ∈ G, note that the pair ρ(Lg ), ρ(Rg ) has the properties (1-3) listed above. So let θ(g) be the element of H obtained from this pair. Similarly, the inverse of θ is determined by the inverse of ρ. By Claim 3.6, θ preserves the composition operation. The identity of G is the only element g ∈ G such that Lg = NG . Since ρ is an isomorphism of the structures, θ(1G ) = 1H , and θ and θ −1 are continuous at 1. So θ is a topological isomorphism. 4. Hardness result for isomorphism of profinite groups Graph isomorphism is complete for S∞ -orbit equivalence relations. We now consider the converse problem of Borel reducing graph isomorphism to isomorphism on a Borel class of nonarchimedean groups. We first consider the case of discrete groups. Essentially by a result of Mekler [7, Section 2] discussed in more detail below, graph isomorphism is Borel reducible to isomorphism of countable groups. Given a discrete group G with domain ω, the left translation action of G on itself induces a topological isomorphism of G with a discrete subgroup of S∞ . Hence graph isomorphism is Borel reducible to isomorphism of discrete, and hence of locally compact, subgroups of S∞ . We now show a similar hardness result for the compact non-archimedean groups (equivalently, the separable profinite groups). Recall that a group G is step 2 nilpotent (nilpotent-2 for short) if it satisfies the law [[x, y], z] = 1.


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Equivalently, the commutator subgroup is contained in the center. For a prime p, the group of unitriangular matrices   1 a c UT33 (Z/pZ) = 0 1 b  : a, b, c ∈ Z/pZ 0 0 1 is an example of a nilpotent-2 group of exponent p. 4.1. Completion. We need some preliminaries on the completion of a group G with respect to a system of subgroups of finite index. We follow [9, Section 3.2]. Let V be a set of normal subgroups of finite index in G such that U, V ∈ V implies that there is W ∈ V with W ⊆ U ∩ V . We can turn G into a topological group by declaring V a basis of neighbourhoods of the identity. In other words, M ⊆ G is open if for each x ∈ M there is U ∈ V such that xU ⊆ M . The completion of G with respect to V is the inverse limit GV = lim G/U, ←− U ∈V

where V is ordered under inclusion and the inverse system is equipped with the natural maps: for U ⊆ V , the map pU,V : G/U → G/V is given by gU 7→ gVQ . The inverse limit can be seen as a closed subgroup of the direct product U ∈V G/U (where each group G/U carries the discrete topology), consisting of the functions α such that pU,V (α(gU )) = gV for each g. Note that the map g 7→ (gU )U ∈V isTa continuous homomorphism G → GV with dense image; it is injective iff V = {1}. b instead If the set V is understood from the context, we will usually write G of GV . 4.2. Review of Mekler’s construction. Fix an odd prime p. The main construction in Mekler [7, Section 2] associates to each symmetric and irreflexive graph A a nilpotent-2 exponent-p group G(A) in such a way that isomorphic graphs yield isomorphic groups. In the countable case, the map G is Borel when viewed as a map from the Polish space of countable graphs to the space of countable groups. To recover A from G(A), Mekler uses a technical restriction on the given graphs. Definition 4.1. A symmetric and irreflexive graph is called nice if it has no triangles, no squares, and for each pair of distinct vertices x, y, there is a vertex z joined to x and not to y. Mekler [7] proves that a nice graph A can be interpreted in G(A) using first-order formulas without parameters. (See [5, Ch. 5] for background on interpretations.) In particular, for nice graphs A, B we have A ∼ = B iff G(A) ∼ = G(B). Since isomorphism of nice graphs is Borel complete for S∞ -orbit equivalence relations, so is isomorphism of countable nilpotent-2 exponent p groups. For an alternative write-up of Mekler’s construction see [5, A.3]. In the following all graphs will be symmetric, irreflexive, and have domain ω. Such a graph is thus given by its set of edges A ⊆ {(r, s) : r < s}. We write rAs (or simply rs if A is understood) for (r, s) ∈ A.

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Let F be the free nilpotent-2 exponent-p group on free generators x0 , x1 , . . .. For r 6= s we write xr,s = [xr , xs ]. As noted in [7], the centre Z(F ) of F is an elementary abelian p-group (so an Fp vector space) with basis xr,s for r < s. Given a graph A, Mekler sets G(A) = F/hxr,s : rAsi. In particular F = G(∅). The centre Z = Z(G(A)) is an abelian group of exponent p freely generated by the xr,s such that ¬rAs. Also G(A)/Z is an abelian group of exponent p freely generated by the Zxi . (Intuitively, when defining G(A) as a quotient of F , exactly the commutators xr,s such that rAs are deleted. We make sure that no vertices are deleted.) Lemma 4.2 (Normal form for G(A), [7, 5]). Every element c of Z can be Q β uniquely written in the form (r,s)∈L xr,sr,s where L ⊆ ω × ω is a finite set of pairs (r, si with r < s and ¬rAs, and 0 < βr,s < p. Every elementQof G(A) can be uniquely written in the form c · v where cQ∈ Z, and v = i∈D xαi i , for D ⊆ ω finite and 0 < αi < p. (The product αi i∈D xi is interpreted along the indices in ascending order.) 4.3. Hardness result for isomorphism of profinite groups. The following first appeared in preprint form in [8]. Theorem 4.3. Let p ≥ 3 be a prime. Any S∞ orbit equivalence relation can be Borel reduced to isomorphism between profinite nilpotent-2 groups of exponent p. We note that isomorphism on the class of abelian compact subgroups of S∞ is not Borel–above graph isomorphism as shown in [8]. So in a sense the class of nilpotent-2 groups of fixed exponent p is the smallest possible. Proof. The proof is based on Mekler’s, replacing the groups G(A) he defined b by their completions G(A) with respect to a suitable basis of neighbourhoods of the identity. Given a graph A, let Rn be the normal subgroup of G(A) generated by the xi , i ≥ n. Note that G(A)/Rn is a finitely generated b nilpotent torsion group, and hence finite. Let G(A) be the completion of G(A) with respect to the set V = {Rn : n ∈ ω} (see Subsection 4.1). By T b Lemma 4.2 we have n Rn = {1}, so G(A) embeds into G(A). In set theory one inductively defines 0 = ∅ and n = {0, . . . , n − 1} to obtain the natural numbers; this will save on notation here. A set of coset representatives for G(A)/Rn is given by the c · v as in Lemma 4.2, where b D ⊆ n and E ⊆ n × n. The completion G(A) of G(A) with respect to the Q Rn consists of the maps ρ ∈ n G(A)/Rn such that ρ(gRn+1 ) = gRn for each n ∈ ω and g ∈ G(A). If ρ(gRn+1 ) = hRn where h = c · v is a coset representative, then we can define a coset representative c′ · v ′ for gRn+1 as follows: we obtain c′ from c by potentially appending to c factors involving the xr,n for r < n, and v ′ from v by potentially appending a factor xαnn . So we can view ρ as given by multiplying two formal infinite products:


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b b Lemma 4.4 (Normal form for G(A)). Every c ∈ Z(G(A)) can be written Q βr,s uniquely in the form (r,s)∈L xr,s where L ⊆ ω × ω is a set of pairs (r, si with r < s, ¬rAs, and 0 < βr,s < p. b Every element of G(A) can be written uniquely in the form c · v, where Q αi b v = i∈D xi , c ∈ Z(G(A)), D ⊆ ω, and 0 < αi < p (the product is taken along ascending indices). b We can define the infinite products above explicitly as limits in G(A). We b view G(A) as embedded into G(A). Given formal products as above, let Q Q vm = i∈D∩m xαi i and cm = (r,s)∈L∩m×m xβr,srs . For k ≥ n we have vk−1 vn ∈ Rn and c−1 k cn ∈ Rn . So v = limn vn and b c = limn cn exist in G(A) and equal the values of the formal products as defined above. Each nilpotent-2 group satisfies the distributive law [x, yz] = [x, y][x, z]. This implies that [xαr , xβs ] = xαβ r,s . The following lemma generalises to infinite products the expression for commutators that were obtained using these identities in [7, p. 784] (and also in [5, proof of Lemma A.3.4]). b Lemma 4.5 (Commutators). Let D, E ⊆ ω. The following holds in G(A). Y Y Y [ xr αr , xs β s ] = xαr,sr βs −αs βr r∈D

r∈D, s∈E, r